Sometimes when I’m really tired in the morning I get out of bed like this, it’s the path of leadt resistance.

A practical application is for example in probability theory (or anywhere that deals with measures) such as this question:

If we generate a random real number from 0 to 1, what is the probability that it is rational?

Because we know that the continuum is so much larger in a sense than the set of rationals, we can answer this confidently and say the probability is zero, even though it is theoretically possible for us to get a rational number.

Statistics deals with similar scenarios quite frequently, and without it we wouldn’t have the modern scientific method.

There are infinitely many rational numbers between any two integers (or any two rationals), yet the rationals are still countable, so this reasoning doesn’t hold.

The only simple intuition for the uncountability of the reals I know of is Cantor’s diagonal argument.

I think aliexpress EMF paint has a good chance of working. All you need is some kind of conductor (usually graphite, that’s literally just pencil lead) mixed with some kind of adhesive. I’m not sure how the adhesive doesn’t stop electricity flow (maybe there’s just so much graphite it doesn’t matter), anyways you can buy graphite spray in any electronics hobby store, a paint is just that with a different consistency.

If you make a room’s walls and ceiling/floor electrically conductive (with paint or tin foil or wire mesh), you create a faraday cage. And as long as the gaps are small enough (proportional to the wavelength of the signal you want to block), nothing in those frequencies is getting in or out. A Faraday bag for your phone might be cheaper, though.

A game developer can have massive blind spots about their game, feedback can elevate the artist’s vision if responded to critically.

Are tedium and difficulty the same thing?

Difficulty and fun are orthogonal, one doesn’t exclude the other. Look at something like Celeste, it gets brutally difficult in farewell but never feels unfair.

A difficult game should motivate the player to get better, and that can happen while still remaining a fun experience. Nine Sols has a difficult combat system, but the combat feels fluid and fast-paced. Learning how to parry an enemy’s attacks and get in counter hits feels amazing, also you get lots of meaningful upgrades and rewards from everywhere.

Meanwhile in silksong you have a character with insane horizontal movement, then you repeatedly get pushed through narrow vertical platforming sections with flying enemies above, spikes on the sides. Killing a bell guy by agonizingly hitting them 1 hit at a time before they dash away out of reach again is not a fun combat paradigm and doesn’t create the power fantasy Hollow Knight excelled at.

The boss fights and >!Mt.Fay!< are fantastic but exploring the map just feels frustrating, especially when 90% of zones don’t drop any rosaries.

I think for straight 4/4 (so not felt in half time) 140 and above is definitely fast, considering the vast majority of music falls in the 110-130BPM range.

Can confirm it works fine with wine / proton

It is just a definition, but it’s the only definition of the complex exponential function which is well behaved and is equal to the real variable function on the real line.

Also, every identity about analytical functions on the real line also holds for the respective complex function (excluding things that require ordering). They should have probably explained it.